non-isomorphic completions of ℚℚ\mathbb{Q}

Proof. Let’s assume the existence of a field isomorphismf:ℝ→ℚpnormal-:fnormal-→ℝsubscriptℚpf:\,\mathbb{R}\to\mathbb{Q}_{p} for some positiveprime numberppp. If we denote f⁢(p)=afpaf(\sqrt{p})=a, then we obtain